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Basic 8 · Mathematics · Union and Intersection of Sets

Sets of Factors, Union and Intersection, and Powers of Ten

In this topic you will use sets of factors and multiples to find the HCF and LCM, solve real-life problems involving union and intersection, and multiply and divide by powers of 10 including decimals and benchmark fractions.

By the end of these notes you should be able to:

  • Use sets of factors to find common factors and the HCF
  • Solve problems involving union and intersection of sets
  • Multiply and divide by powers of 10

What you already know

  • Factors, multiples and perfect squares

Copy everything that has a blue line on its left into your notebook. The other parts are for reading.

Lesson 1

Strand
Number
Sub-strand
Union and Intersection of Sets
Indicator
B8.1.1.2.2 – Use the knowledge on sets and sets of factors of numbers to solve real life problems involving union and intersection

Introduction

Sets give us a neat way to organise factors and multiples, and to see what numbers have in common.

1. Sets and Notation

Set vocabulary
TermSymbolMeaning
set{ }a collection of items
member∈is an element of
intersection∩items common to both sets
union∪all items in either set
empty set{ } or ∅a set with no members

2. Sets of Factors

Find common factors

Let A = factors of 12 and B = factors of 18. Find A ∩ B and the HCF.

  1. A = {1, 2, 3, 4, 6, 12}.
  2. B = {1, 2, 3, 6, 9, 18}.
  3. A ∩ B = {1, 2, 3, 6}.

Answer: The HCF is 6, the largest member of A ∩ B.

3. Sets of Multiples

Find the LCM

Let M = multiples of 4 and N = multiples of 6, up to 40. Find M ∩ N and the LCM.

  1. M = {4, 8, 12, 16, 20, 24, 28, 32, 36, 40}.
  2. N = {6, 12, 18, 24, 30, 36}.
  3. M ∩ N = {12, 24, 36}.

Answer: The LCM is 12, the smallest member of M ∩ N.

A Venn diagram drawn on a chalkboard
Figure 1: Venn diagram Draw a Venn diagram of the factors of 12 and 18.

4. Venn Diagrams

A Venn diagram shows sets as overlapping circles. Members common to both sets go in the overlap, and the others go in the outer parts. In the example above, 1, 2, 3 and 6 go in the overlap, while 4 and 12 go only in A, and 9 and 18 go only in B. The picture makes the relationship between the sets clear.

New words

intersection
Members common to two sets.
union
All members of two sets together.
Venn diagram
Overlapping circles showing sets.
LCM
The lowest common multiple.

Summary

  • The intersection of sets of factors gives common factors, and the largest is the HCF.
  • The intersection of sets of multiples gives common multiples, and the smallest is the LCM.
  • Venn diagrams show the relationships between sets.

Class activity: Factor Venn Diagrams

  1. Choose two numbers below 50.
  2. List their factors as sets.
  3. Draw a Venn diagram and find the HCF.

You need: Bottle tops, Two raffia hoops, Number cards

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. The symbol ∩ means…

    • A. intersection
    • B. union
    • C. member
    • D. empty
    Show answer
    Answer: intersectionCommon items.

  2. 2. The HCF of 16 and 24 is…

    • A. 8
    • B. 4
    • C. 2
    • D. 48
    Show answer
    Answer: 8Largest common factor.

  3. 3. A ∪ B contains all the members of both sets.

    • A. True
    • B. False
    Show answer
    Answer: TrueUnion.

  4. 4. The LCM of 3 and 5 is ______.

    Show answer
    Answer: 15Smallest common multiple.

  5. 5. List the factors of 20 and 30, and find their intersection.

    Show answer
    Answer: {1, 2, 4, 5, 10, 20} and {1, 2, 3, 5, 6, 10, 15, 30}; intersection {1, 2, 5, 10}.Sets.

  6. 6. Find the LCM of 6, 8 and 12 using multiples.

    Show answer
    Answer: 24Common multiple.

Lesson 2

Strand
Number
Sub-strand
Union and Intersection of Sets
Indicator
B8.1.1.2.2 – Use the knowledge on sets and sets of factors of numbers to solve real life problems involving union and intersection

Introduction

A class survey asks who plays football and who plays volleyball. Some learners play both. Sets help us to count them without counting anyone twice.

1. The Formula

For two sets A and B: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Here n(A) means the number of members in set A. We subtract the intersection because those members were counted twice.

2. Worked Problem

Sports survey

In a class of 40 learners, 25 play football, 18 play volleyball and 8 play both. How many play at least one game? How many play neither?

  1. n(F ∪ V) = 25 + 18 − 8 = 35.
  2. Neither: 40 − 35 = 5.
  3. Football only: 25 − 8 = 17; volleyball only: 18 − 8 = 10.

Answer: 35 play at least one game and 5 play neither.

Students playing volleyball at school
Figure 2: School sports Draw a Venn diagram for the sports survey.

3. Using a Venn Diagram

Fill in a Venn diagram from the inside out. First write the intersection (8). Then subtract to find "only" parts (17 and 10). Finally, add the number outside the circles (5) for the learners in neither set. All the parts must add up to the total: 17 + 8 + 10 + 5 = 40.

4. Finding a Missing Value

Find x

In a group of 30 traders, 20 sell tomatoes, 15 sell onions and each trader sells at least one of them. How many sell both?

  1. Let x be the number selling both.
  2. 20 + 15 − x = 30.
  3. 35 − x = 30.

Answer: x = 5

New words

survey
A way of collecting information from people.
n(A)
The number of members in set A.
neither
Not one nor the other.
at least one
One or more.

Summary

  • n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
  • Fill a Venn diagram from the intersection outwards.
  • All parts must add up to the total.

Class activity: Class Survey

  1. Survey your class on two choices, such as subjects or sports.
  2. Record the numbers in a Venn diagram.
  3. Write the results using set notation.

You need: Bottle tops, Two raffia hoops, Number cards

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. In a Venn diagram, members common to both sets go in the…

    • A. overlap
    • B. outside
    • C. corner
    • D. title
    Show answer
    Answer: overlapIntersection.

  2. 2. n(A) = 12, n(B) = 9, n(A ∩ B) = 4. n(A ∪ B) =

    • A. 17
    • B. 21
    • C. 25
    • D. 13
    Show answer
    Answer: 1712 + 9 − 4.

  3. 3. All parts of a Venn diagram must add up to the total.

    • A. True
    • B. False
    Show answer
    Answer: TrueCheck.

  4. 4. A way of collecting information from people is a ______.

    Show answer
    Answer: surveyQuestionnaire.

  5. 5. In a class of 35, 20 like Maths, 18 like Science and 10 like both. How many like neither?

    Show answer
    Answer: 7 (35 − 28)Formula.

  6. 6. 50 people were asked: 32 read newspapers and 27 listen to radio. All do at least one. How many do both?

    Show answer
    Answer: 9 (32 + 27 − 50)Find x.

Lesson 3

Strand
Number
Sub-strand
Decimals
Indicator
B8.1.2.1.1 – Multiply and divide by power of 10 including decimals and the benchmark fractions

Introduction

Multiplying by 10, 100 or 0.01 is quick once you understand how digits shift in place value.

1. Review

Effects of powers of ten
OperationEffectExample
× 10ⁿdigits move n places left3.72 × 10³ = 3,720
÷ 10ⁿdigits move n places right45.6 ÷ 10² = 0.456
× 0.1 or 1/10same as ÷ 1052 × 0.1 = 5.2
× 0.01 or 1/100same as ÷ 10052 × 0.01 = 0.52
÷ 0.1same as × 105.2 ÷ 0.1 = 52
÷ 0.01same as × 1005.2 ÷ 0.01 = 520

2. Benchmark Fractions

Multiplying by ½ is the same as dividing by 2, and multiplying by ¼ is the same as dividing by 4. Dividing by ½ is the same as multiplying by 2. So 48 ÷ ½ = 96. This is because there are 96 halves in 48.

Calculate

Find 0.45 × 1,000, 36 ÷ 0.01 and 80 ÷ ¼.

  1. 0.45 × 1,000 = 450.
  2. 36 ÷ 0.01 = 36 × 100 = 3,600.
  3. 80 ÷ ¼ = 80 × 4 = 320.

Answer: 450; 3,600; 320

A metre rule with centimetre markings
Figure 3: Metre rule Convert five lengths between metres, centimetres and millimetres.

3. Unit Conversions

Metric units are based on powers of ten. 1 km = 1,000 m, 1 m = 100 cm, 1 cm = 10 mm, 1 kg = 1,000 g and 1 litre = 1,000 ml. So 3.5 km = 3,500 m, 750 g = 0.75 kg and 2.4 litres = 2,400 ml.

4. Multiplying in Standard Form

To multiply numbers in standard form, multiply the first numbers and add the indices: (2 × 10³) × (4 × 10⁵) = 8 × 10⁸. To divide, divide the first numbers and subtract the indices: (9 × 10⁷) ÷ (3 × 10²) = 3 × 10⁵.

New words

power of ten
10 multiplied by itself a number of times.
benchmark fraction
A common fraction such as ½ or ¼.
metric unit
A unit based on powers of ten.
convert
To change from one unit to another.

Summary

  • Multiplying moves digits left, and dividing moves them right.
  • Multiplying by 0.1 is dividing by 10, and dividing by ½ is multiplying by 2.
  • Metric conversions and standard form use powers of ten.

Class activity: Conversion Race

  1. Convert ten measurements between metric units.
  2. Check using multiplication or division by powers of ten.
  3. Time yourselves and try to improve.

You need: Fraction strips, Bottle tops, Graph paper

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. 4.5 × 100 =

    • A. 450
    • B. 45
    • C. 4,500
    • D. 0.45
    Show answer
    Answer: 450Two places left.

  2. 2. 60 ÷ 0.1 =

    • A. 600
    • B. 6
    • C. 60
    • D. 0.6
    Show answer
    Answer: 600× 10.

  3. 3. Dividing by ½ is the same as multiplying by 2.

    • A. True
    • B. False
    Show answer
    Answer: TrueInverse.

  4. 4. 2.5 km = ______ m.

    Show answer
    Answer: 2,500× 1,000.

  5. 5. Calculate 0.72 ÷ 0.01.

    Show answer
    Answer: 72× 100.

  6. 6. Calculate (3 × 10⁴) × (2 × 10³).

    Show answer
    Answer: 6 × 10⁷Add indices.

Homework

  1. Draw Venn diagrams for two survey problems.
  2. Convert ten measurements between units using powers of ten.
Sets of Factors, Union and Intersection, and Powers of Ten – Basic 8 Mathematics Notes | sirbus